# 10.2 Permutations and applications  (Page 2/2)

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## Permutation with repetition

When order matters and an object can be chosen more than once then the number of

permutations is:

${n}^{r}$

where $n$ is the number of objects from which you can choose and $r$ is the number to be chosen.

For example, if you have the letters A, B, C, and D and you wish to discover the number of ways of arranging them in three letter patterns (trigrams) you find that there are ${4}^{3}$ or 64 ways. This is because for the first slot you can choose any of the four values, for the second slot you can choose any of the four, and for the final slot you can choose any of the four letters. Multiplying them together gives the total.

## The binomial theorem

In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. Its simplest version reads

${\left(x+y\right)}^{n}=\sum _{k=0}^{n}\left(\genfrac{}{}{0pt}{}{n}{k}\right){x}^{k}{y}^{n-k}$

Whenever $n$ is a positive integer, the numbers

$\left(\genfrac{}{}{0pt}{}{n}{k}\right)=\frac{n!}{k!\left(n-k\right)!}$

are the binomial coefficients (the coefficients in front of the powers).

For example, here are the cases n = 2, n = 3 and n = 4:

$\begin{array}{c}\hfill {\left(x+y\right)}^{2}={x}^{2}+\mathbf{2}xy+{y}^{2}\\ \hfill {\left(x+y\right)}^{3}={x}^{3}+\mathbf{3}{x}^{2}y+\mathbf{3}x{y}^{2}+{y}^{3}\\ \hfill {\left(x+y\right)}^{4}={x}^{4}+\mathbf{4}{x}^{3}y+\mathbf{6}{x}^{2}{y}^{2}+\mathbf{4}x{y}^{3}+{y}^{4}\end{array}$

The coefficients form a triangle, where each number is the sum of the two numbers above it:

This formula, and the triangular arrangement of the binomial coefficients, are often attributed to Blaise Pascal who described them in the 17th century. It was, however, known to the Chinese mathematician Yang Hui in the 13th century, the earlier Persian mathematician Omar KhayyÃ¡m in the 11th century, and the even earlier Indian mathematician Pingala in the 3rd century BC.

The number plate on a car consists of any 3 letters of the alphabet (excluding the vowels and 'Q'), followed by any 3 digits (0 to 9). For a car chosen at random, what is the probability that the number plate starts with a 'Y' and ends with an odd digit?

1. The number plate starts with a 'Y', so there is only 1 choice for the first letter, and ends with an even digit, so there are 5 choices for the last digit (1, 3, 5, 7, 9).

2. Use the counting principle. For each of the other letters, there are 20 possible choices (26 in the alphabet, minus 5 vowels and 'Q') and 10 possible choices for each of the other digits.

Number of events = $1×20×20×10×10×5=200\phantom{\rule{0.277778em}{0ex}}000$

3. Use the counting principle. This time, the first letter and last digit can be anything.

Total number of choices = $20×20×20×10×10×10=8\phantom{\rule{0.277778em}{0ex}}000\phantom{\rule{0.277778em}{0ex}}000$

4. The probability is the number of events we are counting, divided by the total number of choices.

Probability = $\frac{200\phantom{\rule{0.277778em}{0ex}}000}{8\phantom{\rule{0.277778em}{0ex}}000\phantom{\rule{0.277778em}{0ex}}000}=\frac{1}{40}=0,025$

Show that

$\frac{n!}{\left(n-1\right)!}=n$
1. Method 1: Expand the factorial notation.

$\frac{n!}{\left(n-1\right)!}=\frac{n×\left(n-1\right)×\left(n-2\right)×...×2×1}{\left(n-1\right)×\left(n-2\right)×...×2×1}$

Cancelling the common factor of $\left(n-1\right)×\left(n-2\right)×...×2×1$ on the top and bottom leaves $n$ .

So $\frac{n!}{\left(n-1\right)!}=n$

2. Method 2: We know that $P\left(n,r\right)=\frac{n!}{\left(n-r\right)!}$ is the number of permutations of $r$ objects, taken from a pool of $n$ objects. In this case, $r=1$ . To choose 1 object from $n$ objects, there are $n$ choices.

So $\frac{n!}{\left(n-1\right)!}=n$

## Exercises

1. Tshepo and Sally go to a restaurant, where the menu is:
 Starter Main Course Dessert Chicken wings Beef burger Chocolate ice cream Mushroom soup Chicken burger Strawberry ice cream Greek salad Chicken curry Apple crumble Lamb curry Chocolate mousse Vegetable lasagne
1. How many different combinations (of starter, main course, and dessert) can Tshepo have?
2. Sally doesn't like chicken. How many different combinations can she have?
2. Four coins are thrown, and the outcomes recorded. How many different ways are there of getting three heads? First write out the possibilities, and then use the formula for combinations.
3. The answers in a multiple choice test can be A, B, C, D, or E. In a test of 12 questions, how many different ways are there of answering the test?
4. A girl has 4 dresses, 2 necklaces, and 3 handbags.
1. How many different choices of outfit (dress, necklace and handbag) does she have?
2. She now buys two pairs of shoes. How many choices of outfit (dress, necklace, handbag and shoes) does she now have?
5. In a soccer tournament of 9 teams, every team plays every other team.
1. How many matches are there in the tournament?
2. If there are 5 boys' teams and 4 girls' teams, what is the probability that the first match will be played between 2 girls' teams?
6. The letters of the word 'BLUE' are rearranged randomly. How many new words (a word is any combination of letters) can be made?
7. The letters of the word 'CHEMISTRY' are arranged randomly to form a new word. What is the probability that the word will start and end with a vowel?
8. There are 2 History classes, 5 Accounting classes, and 4 Mathematics classes at school. Luke wants to do all three subjects. How many possible combinations of classes are there?
9. A school netball team has 8 members. How many ways are there to choose a captain, vice-captain, and reserve?
10. A class has 15 boys and 10 girls. A debating team of 4 boys and 6 girls must be chosen. How many ways can this be done?
11. A secret pin number is 3 characters long, and can use any digit (0 to 9) or any letter of the alphabet. Repeated characters are allowed. How many possible combinations are there?

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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